Hemispherical electron energy analyzer
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A hemispherical electron energy analyzer or hemispherical deflection analyzer is a type of electron energy spectrometer generally used for applications where high energy resolution is needed—different varieties of
electron spectroscopy Electron spectroscopy refers to a group formed by techniques based on the analysis of the energies of emitted electrons such as photoelectrons and Auger electrons. This group includes X-ray photoelectron spectroscopy (XPS), which also known as Ele ...
such as
angle-resolved photoemission spectroscopy Angle-resolved photoemission spectroscopy (ARPES) is an experimental technique used in condensed matter physics to probe the allowed energies and momenta of the electrons in a material, usually a crystalline solid. It is based on the photoelec ...
(ARPES),
X-ray photoelectron spectroscopy X-ray photoelectron spectroscopy (XPS) is a surface-sensitive quantitative spectroscopic technique based on the photoelectric effect that can identify the elements that exist within a material (elemental composition) or are covering its surface, ...
(XPS) and
Auger electron spectroscopy file:HD.6C.037 (11856519893).jpg, A Hanford Site, Hanford scientist uses an Auger electron spectrometer to determine the elemental composition of surfaces. Auger electron spectroscopy (AES; pronounced in French) is a common analytical technique us ...
(AES) or in imaging applications such as
photoemission electron microscopy Photoemission electron microscopy (PEEM, also called photoelectron microscopy, PEM) is a type of electron microscopy that utilizes local variations in electron emission to generate image contrast. The excitation is usually produced by ultraviolet l ...
(PEEM) and low-energy electron microscopy (LEEM). It consists of two concentric conductive hemispheres that serve as electrodes that bend the trajectories of the electrons entering a narrow slit at one end so that their final radii depend on their kinetic energy. The analyzer, therefore, provides a mapping from kinetic energies to positions on a detector.


Function

An ideal hemispherical analyzer consists of two concentric hemispherical electrodes (inner and outer hemispheres) of radii R_ and R_ held at proper voltages. In such a system, the electrons are linearly dispersed, depending on their kinetic energy, along the direction connecting the entrance and the exit slit, while the electrons with the same energy are first-order focused. When two voltages, V_ and V_, are applied to the inner and outer hemispheres, respectively, the electric potential in the region between the two electrodes follows from the
Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...
: : V(r)= - \left frac\rightcdot\frac + const. The electric field, pointing radially from the center of the hemispheres out, has the familiar planetary motion 1/r^2 form : , \mathbf(r), = - \left frac\rightcdot\frac The voltages are set in such a way that the electrons with kinetic energy E_k equal to the so-called ''pass energy'' E_\textrm follow a circular trajectory of radius R_ = \tfrac(R_1 + R_2). The
centripetal force A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
along the path is imposed by the electric field -e \mathbf(r). With this in mind, :V (r) = \frac\frac+const. The potential difference between the two hemispheres needs to be :V_-V_=\frac\left(\frac-\frac\right)E_\textrm. A single pointlike detector at radius R_\textrm on the other side of the hemispheres will register only the electrons of a single kinetic energy. The detection can, however, be parallelized because of nearly linear dependence of the final radii on the kinetic energy. In the past, several discrete electron detectors ( channeltrons) were used, but now microchannel plates with phosphorescent screens and camera detection prevail. In general, these trajectories are described in polar coordinates r, \varphi for the plane of the
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
for electrons impinging at an angle \alpha with respect to the normal to the entrance, and for the initial radii r_0 \equiv r(\varphi=0) to account for the finite aperture and slit widths (typically 0.1 to 5 mm): : r(\varphi)=r_0\,\left
right Rights are law, legal, social, or ethics, ethical principles of Liberty, freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convent ...
:where : c^=R_\left tfracr_0\cos^\alpha-2\left(r_0-R_\right)\right As can be seen in the pictures of calculated electron trajectories, the finite slit width maps directly into energy detection channels (thus confusing the real energy spread with the beam width). The angular spread, while also worsening the energy resolution, shows some focusing as the equal negative and positive deviations map to the same final spot. When these deviations from the central trajectory are expressed in terms of the small parameters \varepsilon, \sigma defined as E_k=(1+\varepsilon)E_\textrm, r_0=(1+\sigma)R_\textrm, and having in mind that \alpha itself is small (of the order of 1°), the final radius of the electron's trajectory, r(\pi), can be expressed as :r_\pi\approx R_\textrm(1+2\varepsilon-\sigma-2\alpha^2+2\varepsilon^2-6\alpha^2\varepsilon). If electrons of one fixed energy E_k were entering the analyzer through a slit that is w wide, they would be imaged on the other end of the analyzer as a spot w wide. If their maximal angular spread at the entrance is \alpha, an additional width of 2R_P\,\alpha^2 is acquired, and a single energy channel is smeared over , \Delta r_\pi, _=w+2R_P\,\alpha^2 at the detector side. But there, this additional width is interpreted as energy dispersion, which is, to the first order, , \Delta r_\pi, _\varepsilon = 2R_P\,\Delta E/E_P. It follows that the instrumental energy resolution, given as a function of the width of the slit, w, and the maximal incidence angle, \alpha, of the incoming photoelectrons, which is itself dependent on the width of the aperture and slit, is : \Delta E=E_\textrm\left(\frac+\alpha ^2\right) . The analyzer resolution improves with increasing R_\textrm. However, technical problems related to the size of the analyzer put a limit on its actual value, and most analyzers have it in the range of 100–200 mm. Lower pass energies E_\textrm also improve the resolution, but then the electron transmission probability is reduced, and the signal-to-noise ratio deteriorates accordingly. The electrostatic lenses in front of the analyzer have two main purposes: they collect and focus the incoming photoelectrons into the entrance slit of the analyzer, and they decelerate the electrons to the range of kinetic energies around E_\textrm{P}, in order to increase the resolution. When acquiring spectra in ''swept'' (or ''scanning'') mode, the voltages of the two hemispheres – and hence the pass energy – are held fixed; at the same time, the voltages applied to the electrostatic lenses are swept in such a way that each channel counts electrons with the selected kinetic energy for the selected amount of time. In order to reduce the acquisition time per spectrum, the so-called ''snapshot'' (or ''fixed'') mode can be used. This mode exploits the relation between the kinetic energy of a photoelectron and its position inside the detector. If the detector energy range is wide enough, and if the photoemission signal collected from all the channels is sufficiently strong, the photoemission spectrum can be obtained in one single shot from the image of the detector.


See also

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Mass spectrometry Mass spectrometry (MS) is an analytical technique that is used to measure the mass-to-charge ratio of ions. The results are presented as a ''mass spectrum'', a plot of intensity as a function of the mass-to-charge ratio. Mass spectrometry is use ...


References

Electron spectroscopy